2 * Single-precision pow function.
4 * Copyright (c) 2017-2019, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
10 #include "math_config.h"
13 POWF_LOG2_POLY_ORDER = 5
16 ULP error: 0.82 (~ 0.5 + relerr*2^24)
17 relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
18 relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
19 relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
22 #define N (1 << POWF_LOG2_TABLE_BITS)
23 #define T __powf_log2_data.tab
24 #define A __powf_log2_data.poly
25 #define OFF 0x3f330000
27 /* Subnormal input is normalized so ix has negative biased exponent.
28 Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */
29 static inline double_t
30 log2_inline (uint32_t ix)
32 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
33 double_t z, r, r2, r4, p, q, y, y0, invc, logc;
34 uint32_t iz, top, tmp;
37 /* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
38 The range is split into N subintervals.
39 The ith subinterval contains z and c is near its center. */
41 i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
42 top = tmp & 0xff800000;
44 k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
47 z = (double_t) asfloat (iz);
49 /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
51 y0 = logc + (double_t) k;
53 /* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */
66 #define N (1 << EXP2F_TABLE_BITS)
67 #define T __exp2f_data.tab
68 #define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
70 /* The output of log2 and thus the input of exp2 is either scaled by N
71 (in case of fast toint intrinsics) or not. The unscaled xd must be
72 in [-1021,1023], sign_bias sets the sign of the result. */
74 exp2_inline (double_t xd, uint32_t sign_bias)
77 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
78 double_t kd, z, r, r2, y, s;
81 # define C __exp2f_data.poly_scaled
82 /* N*x = k + r with r in [-1/2, 1/2] */
83 kd = roundtoint (xd); /* k */
84 ki = converttoint (xd);
86 # define C __exp2f_data.poly
87 # define SHIFT __exp2f_data.shift_scaled
88 /* x = k/N + r with r in [-1/(2N), 1/(2N)] */
89 kd = eval_as_double (xd + SHIFT);
91 kd -= SHIFT; /* k/N */
95 /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
98 t += ski << (52 - EXP2F_TABLE_BITS);
105 return eval_as_float (y);
108 /* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is
109 the bit representation of a non-zero finite floating-point value. */
111 checkint (uint32_t iy)
113 int e = iy >> 23 & 0xff;
118 if (iy & ((1 << (0x7f + 23 - e)) - 1))
120 if (iy & (1 << (0x7f + 23 - e)))
126 zeroinfnan (uint32_t ix)
128 return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
132 powf (float x, float y)
134 uint32_t sign_bias = 0;
139 if (unlikely (ix - 0x00800000 >= 0x7f800000 - 0x00800000 || zeroinfnan (iy)))
141 /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */
142 if (unlikely (zeroinfnan (iy)))
145 return issignalingf_inline (x) ? x + y : 1.0f;
146 if (ix == 0x3f800000)
147 return issignalingf_inline (y) ? x + y : 1.0f;
148 if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000)
150 if (2 * ix == 2 * 0x3f800000)
152 if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
153 return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
156 if (unlikely (zeroinfnan (ix)))
159 if (ix & 0x80000000 && checkint (iy) == 1)
165 if (2 * ix == 0 && iy & 0x80000000)
166 return __math_divzerof (sign_bias);
168 /* Without the barrier some versions of clang hoist the 1/x2 and
169 thus division by zero exception can be signaled spuriously. */
170 return iy & 0x80000000 ? opt_barrier_float (1 / x2) : x2;
172 /* x and y are non-zero finite. */
176 int yint = checkint (iy);
178 return __math_invalidf (x);
180 sign_bias = SIGN_BIAS;
185 /* Normalize subnormal x so exponent becomes negative. */
186 ix = asuint (x * 0x1p23f);
191 double_t logx = log2_inline (ix);
192 double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec. */
193 if (unlikely ((asuint64 (ylogx) >> 47 & 0xffff)
194 >= asuint64 (126.0 * POWF_SCALE) >> 47))
196 /* |y*log(x)| >= 126. */
197 if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
198 /* |x^y| > 0x1.ffffffp127. */
199 return __math_oflowf (sign_bias);
200 if (WANT_ROUNDING && WANT_ERRNO
201 && ylogx > 0x1.fffffffa3aae2p+6 * POWF_SCALE)
202 /* |x^y| > 0x1.fffffep127, check if we round away from 0. */
204 && eval_as_float (1.0f + opt_barrier_float (0x1p-25f)) != 1.0f)
206 && eval_as_float (-1.0f - opt_barrier_float (0x1p-25f))
208 return __math_oflowf (sign_bias);
209 if (ylogx <= -150.0 * POWF_SCALE)
210 return __math_uflowf (sign_bias);
212 if (ylogx < -149.0 * POWF_SCALE)
213 return __math_may_uflowf (sign_bias);
216 return exp2_inline (ylogx, sign_bias);
219 strong_alias (powf, __powf_finite)
220 hidden_alias (powf, __ieee754_powf)